A $2015\times2015$ chessboard is given, the cells of which are painted white and black alternatively so that the corner cells are black. There are $n{}$ [url=https://i.stack.imgur.com/V1kdh.png]L-trominoes[/url] placed on the board, no two of which overlap and which cover all of the black cells. Find the smallest possible value of $n{}$.

Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$ , where $q$ is an arbitrary rational number. [b]a)[/b] Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element. [b]b)[/b] Determine all the possible values for the cardinality of $M_q$

Given is a triangle $ABC$ and two points $D \in AC, E \in BD$ such that $\angle DAE=\angle AED=\angle ABC$. Show that $BE=2CD$ iff $\angle ACB=90^{\circ}$.

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$

The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$. The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$. Show that $BC = BP$ or $AD = BP$.

We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$

Given a rectangle $ABCD$ with $AB> BC$. On the side $CD$, take a point $L$ such that $BL$ and $AC$ are perpendicular. Let $K$ be the intersection point of segments $BL$ and $AC$. It is known that segments $AL$. and $DK$ are perpendicular. Find $\angle ACB.$

Let $({{A}_{n}})_{n=1}^{\infty }$ and $({{a}_{n}})_{n=1}^{\infty }$ be sequences of positive integers. Assume that for each positive integer $x$, there is a unique positive integer $N$ and a unique $N-tuple$ $({{x}_{1}},...,{{x}_{N}})$ such that $0\le {{x}_{k}}\le {{a}_{k}}$ for $k=1,2,...N$, ${{x}_{N}}\ne 0$, and $x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}$. (a) Prove that ${{A}_{k}}=1$ for some $k$; (b) Prove that ${{A}_{k}}={{A}_{j}}\Leftrightarrow k=j$; (c) Prove that if ${{A}_{k}}\le {{A}_{j}}$, then $\left. {{A}_{k}} \right|{{A}_{j}}$.

$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.

The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ . .

Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

Let be a sequence $ \left( s_n\right)_{n\geqslant 0} $ of positive real numbers, with $ s_0 $ being the golden ratio, and defined as $$ s_{n+2}=\frac{1+s_{n+1}}{s_n} . $$ Establish the necessary and sufficient condition under which $ \left( s_n\right)_{n\geqslant 0} $ is convergent.

Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1 : 2 : 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of this chord.

A badminton club consists of $2n$ members who are n couples. The club plans to arrange a round of mixed doubles matches where spouses neither play together nor against each other. Requirements are: $\cdot$ Each pair of members of the same gender meets exactly once as opponents in a mixed doubles match. $\cdot$ Any two members of the opposite gender who are not spouses meet exactly once as partners and also as opponents in a mixed doubles match. Given that $(n,6)=1$, can you arrange a round of mixed doubles matches that meets the above specifications and requirements?

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done. [i]Yudi Satria, Jakarta[/i]

Let $ u,v $ be two endomorphisms of a finite vectorial space that verify the relation $ uv-vu=u. $ Calculate $ u^kv-vu^k $ and show that u is nilpotent.

Let $n \geqslant 3$ be a positive integer. Furthermore, let $x_1, x_2,\ldots, x_n \in [0, 2]$ be real numbers subject to $x_1 + x_2 +\cdots + x_n = 5$. Prove the inequality$$x_1^2 + x_2^2 + \cdots + x_n^2 \leqslant 9.$$When does equality hold? [i](Walther Janous)[/i]

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

Let $ABC$ be a triangle, $d$ a line passing through $A$ and parallel to $BC$. A point $M$ distinct from $A$ is chosen on $d$. $I$ is the incenter of triangle $ABC, K,L$ are the the points of symmetry of $M$ about $IB, IC$. Let $BK$ meet $CL$ at $N$. Prove that $AN$ is tangent to circumcircle of triangle $ABC$. Đỗ Thanh Sơn

$100$ betyárs stand on the Hortobágy plains. Every betyár's field of vision is a $100$ degree angle. After each of them announces the number of other betyárs they see, we compute the sum of these $100$ numbers. What is the largest value this sum can attain?

In the figure, BC  is a diameter of the circle, where $BC=\sqrt{901}, BD=1$, and $DA=16$. If $EC=x$, what is the value of x? [asy]size(2inch); pair O,A,B,C,D,E; B=(0,0); O=(2,0); C=(4,0); D=(.333,1.333); A=(.75,2.67); E=(1.8,2); draw(Arc(O,2,0,360)); draw(B--C--A--B); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$D$",D,W); label("$E$",E,N); label("Figure not drawn to scale",(2,-2.5),S); [/asy]

In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.